# Transfer Matrix formalism

I am trying to apply the transfer matrix formalism to an Ising model problem, and am having some difficulties deriving the correct matrix to use.

The problem is as follows. There is an infinite chain of spins, as shown in the diagram below.

The energy of this configuration is given by:

$$E = -J\sum_{i=1}^{L}(\sigma_{i}\sigma_{i+1}+\sigma_{i}\tau_{i} + \tau_{i}\sigma_{i+1})$$

Where we assume periodic boundary conditions, i.e. $\sigma_{L+1}=\sigma_{1}$, and $J > 0$. The partition function can be written:

$$Z(\beta)=\sum_{\text{configuratons }C}e^{-\beta E(C)} = \sum_{\sigma_{1}=\pm1}\cdots\sum_{\sigma_{L}=\pm1}\dots\sum_{\tau_{1}=\pm1}\dots\sum_{\tau_{L}=\pm 1}e^{-\beta E(\{\sigma_{i}\},\{\tau_{i}\})}$$

We can decompose the energy into sub-functions:

$$E(\sigma_{i},\sigma_{i+1},\tau_{i})=-J\left(\sigma_{i}\sigma_{i+1}+\sigma_{i}\tau_{i}+\tau_{i}\sigma_{i+1}\right)$$

This allows us to write the partition function:

$$Z(\beta)=\sum_{\sigma_{1},\dots,\tau_{1},\dots}\prod_{j=1}^{N}e^{-\beta E(\sigma_{j},\sigma_{j+1},\tau_{j})}$$

I wish to form this as a product of matrices so that I can find the eigenvalues and use the property:

$$Z(\beta) = \operatorname{Tr}(T^{N})$$

However, naively, I would think that $T$ was a $2\times 2\times 2$ tensor, rather than a matrix? How can I construct an appropriate transfer matrix? And how can I do it generally, rather than just for this problem?

• I agree that the way you wrote it it's a 2x2x2 transfer tensor. Also, since the boundary conditions are not periodic, I am not sure you will be able to write it in trace form anyway. – Cyclone Apr 25 '17 at 13:12
• @Cyclone I agree, it was a typo, I will fix now! The boundary conditions are periodic? – Thomas Russell Apr 25 '17 at 13:17
• There are many ways to do that, but probably the easiest is to explicitly sum over the $\tau$ variables (for a fixed realization of the $\sigma$ variables). Note that the variable $\tau_i$ only interacts with the spins $\sigma_i$ and $\sigma_{i+1}$, so all sums over the variables $\tau_i$ factorize, each sum providing an effective interaction between consecutive $\sigma$ spins. Once you have gotten rid of the $\tau$ spins in this way, the transfer matrix is constructed as in the standard Ising chain. – Yvan Velenik Apr 25 '17 at 18:38
• Alternatively, just write a $4\times 4$ transfer matrix allowing you to pass from $(\sigma_i,\tau_i)$ to $(\sigma_{i+1},\tau_{i+1})$. – Yvan Velenik Apr 25 '17 at 18:41
• @YvanVelenik Can you provide an example of what you mean, I've been trying to do this in the way that you describe and am having trouble "factorizing" out the $\tau_{i}$ variable! – Thomas Russell May 24 '17 at 10:45